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3080 results in Probability theory and stochastic processes

Page 73 of 154

  • 5 - Measurability

  • R. M. Dudley, Massachusetts Institute of Technology
  • Book:
    Uniform Central Limit Theorems
    Published online:
    05 June 2014
    Print publication:
    24 February 2014, pp 213-238

  • Summary

    The example after Theorem 3.2 showed that for a continuous distribution function F such as for U[0, 1], the set of all possible functions √n(Fn − F), even for n = 1, is nonseparable in the sup norm, and all its subsets are closed, including those corresponding to nonmeasurable sets of possible values of the observation X1. Therefore, the classical definition of convergence in law, or weak convergence, which works in separable metric spaces, does not work in this case, So, in Chapter 3, functions f* and upper expectations E* were used to get around measurability problems.

    But, in the classical Glivenko–Cantelli theorem, saying that supx |(Fn − F)(x)| → 0 almost surely as n → ∞ for any distribution function F on ℝ and its empirical distribution functions Fn (RAP, Theorem 11.4.2), there is no measurability problem. The supremum is measurable, as it can be restricted to rational x by right-continuity of Fn and F. The collection C of left half-lines (−∞, x] is linearly ordered by inclusion and so has S(C) = 1, and for it, not only the Glivenko–Cantelli theorem but, after suitable formulations (Theorem 1.8 or, less specifically, Chapter 3), the uniform central limit theorem (Donsker property) holds for any probability measure P on the Borel sets of ℝ.

  • Preface to the Second Edition

  • R. M. Dudley, Massachusetts Institute of Technology
  • Book:
    Uniform Central Limit Theorems
    Published online:
    05 June 2014
    Print publication:
    24 February 2014, pp xi-xii

  • Summary

    This book developed out of some topics courses given at M.I.T. and my lectures at the St.-Flour probability summer school in 1982. The material of the book has been expanded and extended considerably since then. At the end of each chapter are some problems and notes on that chapter.

    Starred sections are not cited later in the book except perhaps in other starred sections. The first edition had several double-starred sections in which facts were stated without proofs. This edition has no such sections.

    The following, not proved in the first edition, now are: (i) for Donsker's theorem on the classical empirical process αn := √n(Fn − F), and the Komlós–Major–Tusnády strengthening to give a rate of convergence, the Bretagnolle–Massart proof with specified constants; (ii) Massart's form of the Dvoretzky–Kiefer–Wolfowitz inequality for αn with optimal constant; (iii) Talagrand's generic chaining approach to boundedness of Gaussian processes, which replaces the previous treatment of majorizing measures; (iv) characterization of uniform Glivenko–Cantelli classes of functions (from a paper by Dudley, Giné, and Zinn, but here with a self-contained proof); (v) Giné and Zinn's characterization of uniform Donsker classes of functions; (vi) its consequence that uniformly bounded, suitably measurable classes of functions satisfying Pollard's entropy condition are uniformly Donsker; and (vii) Bousquet, Koltchinskii, and Panchenko's theorem that a convex hull preserves the uniform Donsker property.

Page 73 of 154